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In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable, analogous to the definition that a function between topological spaces is continuous if it preserves the topological structure: the preimage of each open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable.

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Formal definitionEdit

Let   and   be measurable spaces, meaning that   and   are sets equipped with respective  -algebras   and  . A function   is said to be measurable if the preimage of   under   is in   for every  ; i.e.

 

If   is a measurable function, we will write

 

to emphasize the dependency on the  -algebras   and  .

Term usage variationsEdit

The choice of  -algebras in the definition above is sometimes implicit and left up to the context. For example, for  ,  , or other topological spaces, the Borel algebra (containing all the open sets) is a common choice. Some authors define measurable functions as exclusively real-valued ones with respect to the Borel algebra.[1]

If the values of the function lie in an infinite-dimensional vector space, other non-equivalent definitions of measurability, such as weak measurability and Bochner measurability, exist.

Notable classes of measurable functionsEdit

  • Random variables are by definition measurable functions defined on probability spaces.
  • If   and   are Borel spaces, a measurable function   is also called a Borel function. Continuous functions are Borel functions but not all Borel functions are continuous. However, a measurable function is nearly a continuous function; see Luzin's theorem. If a Borel function happens to be a section of some map  , it is called a Borel section.
  • A Lebesgue measurable function is a measurable function  , where   is the  -algebra of Lebesgue measurable sets, and   is the Borel algebra on the complex numbers  . Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated. In the case  ,   is Lebesgue measurable iff   is measurable for all  . This is also equivalent to any of   being measurable for all  , or the preimage of any open set being measurable. Continuous functions, monotone functions, step functions, semicontinuous functions, Riemann-integrable functions, and functions of bounded variation are all Lebesgue measurable.[2] A function   is measurable iff the real and imaginary parts are measurable.

Properties of measurable functionsEdit

  • The sum and product of two complex-valued measurable functions are measurable.[3] So is the quotient, so long as there is no division by zero.[1]
  • If   and   are measurable functions, then so is their composition  .[1]
  • If   and   are measurable functions, their composition   need not be  -measurable unless  . Indeed, two Lebesgue-measurable functions may be constructed in such a way as to make their composition non-Lebesgue-measurable.
  • The (pointwise) supremum, infimum, limit superior, and limit inferior of a sequence (viz., countably many) of real-valued measurable functions are all measurable as well.[1][4]
  • The pointwise limit of a sequence of measurable functions   is measurable, where   is a metric space (endowed with the Borel algebra). This is not true in general if   is non-metrizable. Note that the corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence.[5][6]

Non-measurable functionsEdit

Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to find non-measurable functions.

  • So long as there are non-measurable sets in a measure space, there are non-measurable functions from that space. If   is some measurable space and   is a non-measurable set, i.e. if  , then the indicator function   is non-measurable (where   is equipped with the Borel algebra as usual), since the preimage of the measurable set   is the non-measurable set  . Here   is given by
 
  • Any non-constant function can be made non-measurable by equipping the domain and range with appropriate  -algebras. If   is an arbitrary non-constant, real-valued function, then   is non-measurable if   is equipped with the trivial  -algebra  , since the preimage of any point in the range is some proper, nonempty subset of  , and therefore does not lie in  .

See alsoEdit

NotesEdit

  1. ^ a b c d Strichartz, Robert (2000). The Way of Analysis. Jones and Bartlett. ISBN 0-7637-1497-6.
  2. ^ Carothers, N. L. (2000). Real Analysis. Cambridge University Press. ISBN 0-521-49756-6.
  3. ^ Folland, Gerald B. (1999). Real Analysis: Modern Techniques and their Applications. Wiley. ISBN 0-471-31716-0.
  4. ^ Royden, H. L. (1988). Real Analysis. Prentice Hall. ISBN 0-02-404151-3.
  5. ^ Dudley, R. M. (2002). Real Analysis and Probability (2 ed.). Cambridge University Press. ISBN 0-521-00754-2.
  6. ^ Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite Dimensional Analysis, A Hitchhiker’s Guide (3 ed.). Springer. ISBN 978-3-540-29587-7.

External linksEdit